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In advanced mathematics, a graph is a special data structure made up
of lines (called edges) connected together at endpoints (called
vertices or nodes). This definition of the term graph
is in contrast to a popular definition used in early math classes, where a graph
is simply a plot of a function.

Many computer science and mathematical algorithms make use of graphs. Because
graphs are so important in many algorithms, a data structure for a graph is
equally important. Most languages don’t have graph data structures built
in; indeed, C++ doesn’t. Therefore, when you need a graph data structure,
you either have to code one yourself or make use of a third-party library
offering a graph data structure.

The Boost
library is quickly becoming a de facto standard C++ library, and it includes a
graph data structure that’s easy to use yet powerful. In this article, I
discuss the theory behind graphs, and then we explore the Boost library’s
graph structures.

Grasping Graph Theory

As I already mentioned, a graph is simply a set of vertices (the points)
connected by edges (the lines). When two vertices are connected by an edge,
those two vertices are said to be adjacent. If you enjoy mathematics,
you might like to know that this definition can be described a bit more
technically by stating that a graph consists of a finite set of vertices along
with a finite set of unordered pairs of distinct vertices; you can see that the
pairs of vertices represent the edges of the graph.

Graphs provide a visual way of displaying underlying information. For
example, a large airline serves many cities. The airline’s web site might
provide a map showing cities as dots, and lines connecting the cities
representing flights between those cities. This map could be considered a graph.
Similar graphs might include those showing a trucking company’s delivery
routes between cities, or bus routes between cities.

Although, technically speaking, a graph consists simply of vertices connected
by edges, this limited definition isn’t particularly useful. Often, graphs
are more useful when they have more associated information. For example, a
flight map might also show the flight numbers associated with each flight. In
this representation, a graph consists of vertices, edges, and numbers associated
with the edges. When numbers are associated with edges, such graphs are called weighted graphs.

Certainly, some graphs might need more than one piece of data associated with
each edge. For example, in addition to the flight numbers, the airline map might
show the length of each flight. In terms of computer science, however, such
information would still be considered individual pieces of data in the form of
individual structures or objects associated with each edge, with each structure
containing a flight number and a flight time

Similarly, a graph might need to have information attached to the vertices,
such as city names in the case of the airline map. In mathematics, you can
attach numbers to vertices and use such numbers in calculations.

Another somewhat limiting aspect of a traditional definition of a graph is
that the edges connecting the vertices don’t have directions. In real
life, a system that can be modeled by a graph will likely have directions. For
example, if you travel by train somewhere, you definitely want to feel
comfortable that the tracks you’re riding on from one city to the next are
carrying the train in a single direction! When you add directions to graphs, you
have what mathematicians call a directed graph, or digraph for
short.

Important to graph theory is another type of graph, limited by the
restriction that the graph can be drawn on a sheet of paper without any edges
crossing. Such a graph is called a planar graph. The term planar is used because the graph can be drawn on a single plane. Of
course, this plane might not always be flat in the geometrical sense; the plane
could exist on a curved surface such as the surface of the Earth, in which case
it still maps to two dimensions. That’s getting into another interesting
area of mathematics called topology, however, which we won’t
cover here.

TIP

Fascinated by the concepts of graphs? One of my favorite books on the subject
is Gary Chartrand’s Introductory Graph Theory (Dover
Publications, 1984).